A hyperbolic reflection group is a group generated by reflections in hyperbolic space. We call a Coxeter group hyperbolic if it is infinite, nonaffine, and it has a representation as a discrete, properly acting, hyperbolic reflection group whose Tits' cone consists entirely of vectors with negative norm (see [Bou68] for more details). A hyperbolic reflection group is compact hyperbolic if it is hyperbolic with a compact fundamental region.
Every infinite nonaffine Coxeter group of rank 3 is hyperbolic. There are only 72 hyperbolic groups of rank larger than 3--- for convenience, we number these groups from 1 to 72. Our numbering is essentially arbitrary.
Returns true if, and only if, G is the Coxeter matrix of a hyperbolic Coxeter group.
Returns true if, and only if, G is the Coxeter graph of a hyperbolic Coxeter group.
Returns true if, and only if, G is the Coxeter matrix of a compact hyperbolic Coxeter group.
Returns true if, and only if, G is the Coxeter graph of a compact hyperbolic Coxeter group.
The Coxeter matrix of the ith hyperbolic Coxeter group of rank larger than 3.
The Coxeter graph of the ith hyperbolic Coxeter group of rank larger than 3.
> for i in [1..72] do > if IsCoxeterCompactHyperbolic( HyperbolicCoxeterMatrix(i) ) then > printf "%o, ", i; > end if; > end for; 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,